The Duke was out of favor with the King, and when it came time for the Duke to pass down his land, the King hatched a plan to break up the Duke’s holdings (and thus
the political power of his heirs). The King proclaimed that the Duke must
split all of the land among his heirs such that:

1) The land is split into at least two pieces, and
2) Every piece has the same shape and size, allowing for
rotations and/or reflections of the shape.

What are the largest pieces the Duke can break his duchy into, in
accordance with the King’s edict?

Not so fast. All you proved is that the largest piece isn't 6 squares. You haven't proven that the largest piece is smaller than 6. In fact, the solution uses pieces larger than 6 squares. Give it another try.